\; x+8 x/4 x+6 x dx is equal to

Correct Answer is : x+12log⁡(4sin⁡x+6cos⁡x)+cx+1/2(4 x+6 x)+cx+21log(4sinx+6cosx)+c

Correct Answer is : x+12log⁡(4sin⁡x+6cos⁡x)+cx+1/2(4 x+6 x)+cx+21log(4sinx+6cosx)+c

Test Index

EAMCET Engineering 2007 Solved Paper

Section: Mathematics

Question : 153 of 160

Marks: +1, -0

\int\;\frac{\sin x+8\cos x}{4\sin x+6\cos x} dx

Solution:

Let

I=\int\;\frac{\sin x+8\cos x}{4\sin x+6\cos x} dx

We can write

\;\sin x+8\cos x=A(4\sin x+6\cos x)

\;\;\;+B\;\frac{d}{dx}(4\sin x+6\cos x)

\;\sin x+8\cos x

\;=A(4\sin x+6\cos x)+B(4\cos x-6\sin x)

On equating the coefficient of

\sin x

and

\cos x

, we get

\;1=4A-6B,\;\;8=6A+4B

\;\Rightarrow\;\;A=1,\;B=\frac{1}{2}

\therefore I\;=\int\;\frac{(4\sin x+6\cos x)+\frac{1}{2}(4\cos x-6\sin x)}{4\sin x+6\cos x} dx

\;=\int\left(1+\frac{1}{2}\cdot\frac{4\cos x-6\sin x}{4\sin x+6\cos x}\right) dx

\;=x+\frac{1}{2}\log(4\sin x+6\cos x)+c

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